Asian Journal of Mathematics

Volume 27 (2023)

Number 3

Compactness and rigidity of self-shrinking surfaces

Pages: 301 – 314



Tang-Kai Lee (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher codimensional case. In this note, we use measure-theoretic techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.


mean curvature flow, compactness, rigidity

2010 Mathematics Subject Classification

53A07, 53C42

Received 7 October 2021

Accepted 6 March 2023

Published 7 November 2023